Polynomials with zeros and small norm on curves
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Abstract:
This paper considers the problem of how zeros lying on the boundary of a domain influence the norm of polynomials (under the normalization that their value is fixed at a point). It is shown that $k$ zeros raise the norm by a factor $(1+ck/n)$ (where $n$ is the degree of the polynomial), while $k$ excessive zeros on an arc compared to $n$ times the equilibrium measure raise the norm by a factor $\exp (ck^2/n)$. These bounds are sharp, and they generalize earlier results for the unit circle which are connected to some constructions in number theory. Some related theorems of Andrievskii and Blatt will also be strengthened.References
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Additional Information
- Vilmos Totik
- Affiliation: Bolyai Institute, Analysis Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary – and – Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue, PHY 114, Tampa, Florida 33620-5700
- Email: totik@mail.usf.edu
- Received by editor(s): April 12, 2011
- Published electronically: February 23, 2012
- Additional Notes: The author was supported by ERC grant No. 267055
- Communicated by: Michael T. Lacey
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3531-3539
- MSC (2010): Primary 41A10, 31A15
- DOI: https://doi.org/10.1090/S0002-9939-2012-11223-7
- MathSciNet review: 2929021